Someone wrote open source code that can play or download YouTube videos by pretending to be a user on youtube.com.
They used it to make a user-friendly player in JavaScript. (It probably also needs to pipe some requests through a backend server to fool YouTube into thinking that the requests do come from youtube.com, look up CORS: this is probably where the name comes from)
They figured out how your browser, while showing youtube.com, communicates with YouTube's backend servers to get search results, recommendations etc.
They eventually recreated pretty much the complete functionality of youtube.com (and then some).
So you can go to their site, piped.video, and play YouTube videos in a friendlier interface.
They knew they could be taken down so they encouraged other people to host their own copies of the website, known as “instances”. These operate independently but run source code from the same repository. If one is too overwhelmed with users, they can switch to another. If you have a server, you can host your own.

From a user's perspective, Piped does basically 2 things:

• It acts as a middleman between you and YouTube, so YouTube can't trace your activities back to you.
• It gives you its own front-end free of ads and trackers.